Gauss-Bonnet models with cosmological constant and non zero spatial curvature in $D=4$

In the present paper the possibility of eternal universes in Gauss-Bonnet theories of gravity in four dimensions is analysed. It is shown that, for zero spatial curvature and zero cosmological constant, if the coupling is such that $0<f'(\phi)\leq c \exp(\frac{\sqrt{8}}{\sqrt{10}}\phi)$, then there are solutions that are eternal. Similar conclusions are found when a cosmological constant turned on. These conclusions are not generalized for the case when the spatial curvature is present, but we are able to find some general results about the possible nature of the singularities. The presented results correct some dubious arguments in [54], although the same conclusions are reached. On the other hand, these past results are considerably generalized to a wide class of situations which were not considered in [54].